Posted
by
timothyon Friday March 19, 2010 @08:12AM
from the now-prove-perelman-exists dept.

epee1221 writes "The Clay Mathematics Institute has announced its acceptance of Dr. Grigori Perelman's proof of the Poincaré conjecture and awarded the first Millennium Prize. Poincaré questioned whether there exists a method for determining whether a three-dimensional manifold is a spherical: is there a 3-manifold not homologous to the 3-sphere in which any loop can be gradually shrunk to a single point? The Poincaré conjecture is that there is no such 3-manifold, i.e. any boundless 3-manifold in which the condition holds is homeomorphic to the 3-sphere. A sketch of the proof using language intended for the lay reader is available at Wikipedia."

Ricci flow is an incredibly clever and sophisticated set of techniques. It is a very difficult technique to use and is by no means a "cheat code" for manifold questions. Most obviously, Ricci flow has been used with success to answer some aspects of the geometrization conjecture http://en.wikipedia.org/wiki/Geometrization_conjecture [wikipedia.org] but still leaves a lot. In order to have a truly good understanding of low-dimensional manifolds we are likely going to need some additional technique that has not yet been disc

It's amazing that TFA doesn't mention a thing about whether Perelman will actually accept the prize. What will happen to the prize money if he does not accept? The million dollars disappears into Lichtenstein numbered bank accounts 2718-282 and 3141-519?

I am very happy that they have awarded the price only to him, although he did meet the requirement that the proof should be published in a peer-reviewed journal. I am very happy that they did not included those two Chinese guys who did write down the proof (about 260 pages) and claimed that they had proven the conjecture. Perelman was very upset by this especially that other mathematics did not raise their voice. I hope that Perelman will accept the price. He said (some years ago) that he would only decide when the offer was made, if he would except the price or not.

I am very happy that they have awarded the price only to him, although he did meet the requirement that the proof should be published in a peer-reviewed journal. I am very happy that they did not included those two Chinese guys who did write down the proof (about 260 pages) and claimed that they had proven the conjecture. Perelman was very upset by this especially that other mathematics did not raise their voice. I hope that Perelman will accept the price. He said (some years ago) that he would only decide when the offer was made, if he would except the price or not.

Perelman has famously turned down the fields medal and shunned the world since the whole Yau political saga. Will he take this prize? I hope that he will. I think that the whole Yau trying to take the credit for the proof issue, sullied the entire world for Perelman. Perhaps now that the honour is being fairly directed at him in response to his work, Perelman will be able to re-enter society and enjoy some of the fruits of his labour.

Has anyone had a hard answer as to why he turned down the prizes and medals? The author of "Perfect Rigor" seemed to think that Perelman thought the Fields Medal was beneath him. I don't think hiding away from society did him any good, especially if he's expecting other people to defend him when he seems not willing to do so himself.

He does not want to be defended. As far as I read the controversy, he does not want to fight at all, because he (quite rightfully, imO) thinks that science should not be fought over.Criticism is useful. Politics (Yau, you asshole!) is not.

Perelman is not a normal guy (obvious I realize, but hear me out). People like to subscribe 'normal' motives for behaviour they see as abnormal. I think this is why the idea that the fields medal was rejected as 'beneath' him was put forward. Arrogance is simple to understand. But what did Perelman actually say? "[the prize] was completely irrelevant for me. Everybody understood that if the proof is correct then no other recognition is needed". What Perelman was looking for was recognition for solving the problem. This was more important than the fields medal! What he got instead, was Yau and his cohorts claiming to have "really solved it." In Perelman's mind, political play such as this has no place in mathematics! Worse, his peers were not standing up to a) condemn this behaviour, and b) defend his paper. I think an important missing piece was that Perelman had not been officially recognized as having solved the Poincare conjecture. Now that this had been rectified, perhaps the world will be in enough order for him to rejoin it.

Actually, I'd say it isn't even a linguistic but a cultural problem. The New Yorker employed a Russian guy to explain his reasoning; he is a sort of Russian hermit. Imagine if Tolstoy went up against a person like Yau. I think there would be mutual disgust and bad feeling towards the literary community at large. Such a thing probably happened here.

Why not go along with? How about "moral stance"? Or if that's too abstract, how about this: the man is a bona fides genius. If anyone's got the right to ignore fatuous platitudes, I think his intellectual accomplishment confers the privilege. The better question is this: why do you think everyone needs to conform to your notion of "graciousness"?

Perhaps this was not your intent, but you come across as that annoying neighborhood old lady that wants to see to it that everyone "conform", and gossips behind

I shouldn't have attacked like that. I apologize. Thank you for being so controlled in your response. I will debate better henceforth.

Wow. Thanks, guys (or possibly girls [but who are we kidding?]). I don't mean that sarcastically. Every time I read a comment online, my faith in humanity dips a little lower, but I do appreciate the civilized discourse. It's just not very often someone's response to a counter-argument is, "Yeah, I guess you're right. Sorry." I guess Obama does it every now and then, but there's plenty of political posturing involved. I should probably pay less attention to politics. Maybe mathematics, too.

I can't take credit for finding this. Another Slashdotter was kind enough to link it the last time Perelman came up, but I found this to be very enlightening and illustrative of Perelman's personality as well as the whole Yau controversy. It's an article from the New Yorker co-written by Sylvia Nasar, who wrote the biography of John Nash, A Beautiful Mind. It contains what was at the time the only interview with Grigori Perelman, but I'm not sure if that's still true.

Isn't the moral that you can take credit for anything you like? How about, "Sure, the kind Slashdotter who found it contributed a good thirty or thirty-five percent, but bookmarking and linking to it accounts for at least the other seventy-five percent. It was no small task."

If I were going to go full Yau, I would claim that I discovered this article on the New Yorker's website, and that I am incensed because it is clearly an idea that I came up with first, and if you don't believe me just ask all of my Chinese friends.

From what I have read around here he did not reject Fields Medal. Since they never formally offered it to him. Or did they?"We will give it to you if you take it" is just BS.One more point to Perelman.

Manifold = a surface created by taking pieces of paper and warping them. For example, cylinder is a manifold since it can be formed by attaching the two opposite sides of the paper to each other. If you then attach the two circles at the ends of the cylinder, you get a torus (ie. donut).

Homeomorphic = there's a continuous function mapping points from one object to the other. This means that if two points are close to each other in the first object, they will be close together when the homeomorphism (the function) is used to map the points onto the second object. A square and the surface of a sphere, for example, are not homeomorphic since the square has edges and the sphere doesn't, so the mapping function has to jump somewhere, making it not continuous. Generally, two shapes are homeomorphic if you can deform one into the other (see animation here [wikipedia.org])

Homologous = I don't know how that word got in there. It's not in the Wikipedia article.Simply connected = Any line drawn on the manifold that starts and ends at the same point can be slowly shrunk down to one point without taking any part of it off the manifold. A torus is not simply connected, since you can draw a line going around the cylinder and there's no way to take it off.

As for implications, as far as I can see, it just tells us that lots of things can be deformed into spheres and gives us a simple test for determining if something can.

No, the mercator projection doesn't show either the north or south poles.
They are implied, ie you know where they are supposed to be, but they're not
actually on the map. So the mercator projection doesn't map the full earth sphere, only the earth sphere with two poles missing, which is homeomorphic to a cylinder.

I think the question is easier to understand if you knock everything down a dimension, because then it can actually be visualized. Take the surface of any three-dimensional object that doesn't contain any holes (e.g., a cup, but NOT a coffee mug with a handle). Can the surface be stretched/distorted to be shaped into a sphere? The answer is fairly obviously yes. But is this also true for four-dimensional objects? Stop trying to visualize it; you can't. You have to rely on the math instead. But that, I believe, is the question.

I read the summary and what little mathmatical legs I got were sweapt out from under me. I read "A sketch of the proof using language intended for the lay reader is available at Wikipedia." and my instant reaction was "oh thank you god!"
But when I read the wiki over but couldn't get my head around a one-dimensional circle, and a two-dimensional sphere.
Read some other slashdotters posts and and some other wiki pages, and while I know more about manifolds than I ever

That is the problem that was solved. The crazy thing is that it was proven for all dimensions other than ours in 1982. It took that long to prove the conjecture for the three-dimensional world that we live in. That's wild, no?

Not quite true... a 3-sphere is actually the *surface* of a 4-dimensional sphere. So, not exactly something that lives in our world. In topology, the dimensions refer to the dimensionality of the surface, and not the space the surface lives in (ie: a circle drawn on a piece of paper is a 1-sphere, but the surface it was drawn on is 2-dimensional).

Manifolds don't "live in" any space. Yes, a 3-sphere can be imbedded in R^4 (or R^5 or R^n for any n>3), but for its definition, the 3-sphere does not refer to any ambient space whatsoever. Our world might very well be a 3-sphere...if it were large enough, we'd never know the difference, just like the good old ant-on-a-basketball.

It's really all about classifying shapes. For two dimensional things this is pretty easy, at least as far as the topology goes: you need to know the curvature and "how many holes does it have" and that's it -- this is the whole topologist not knowing a coffee cup from a donut since they both have one hole and hence can be deformed one into the other (note that this is two dimensional because we are considering the 2-dimensional surface on the donut and coffee cup). In dimensions higher than two things start getting trickier because more bizarre configurations become possible. Perelman's work, which actually goes toward proving the rather more far reaching Geometrization Conjecture (due to Thurston), essentially lays out how you can classify all the different (from a topological point of view) shapes of things in three dimensions and higher.

What are the implications? Well, one reasonable question is: what is the topology of the universe like; what shape is the universe? Since the universe is a three dimensional manifold that turns out to be tricky. Perelman's work lays out the groundwork to be able to answer such a question.

It's easier to explain the two-dimensional version, that is the version about surfaces. A mathematical surface is a kind of quilt: it's what you get from stitching together patches, each of which looks like a small piece of the plane. Just like with the quilt, if you bend or deform the surface it still is the same surface. Surfaces are completely "floppy".

Now, most real-life quilts are rectangular and have a boundary where they end, but you can also "close" the quilt by stitching the boundary back onto itself -- what you get is a "closed" surface. For example, you can stitch all the boundary together and get a sphere. Or you can stitch opposite sides together and get a "torus" -- the surface of a doughnut. You can also make more complicated quilts, which look like the joining of several doughnuts, i.e. a doughnut with several holes.

Next, one way that the sphere and doughnut-surface differ is that the latter has a hole. The way we express this is by looping a closed piece of string along the surface. With the sphere you can always slide the piece of string off the surface (we say that the sphere is "simply connected"), but with the torus you can run a loop of string along it in such a way that no deformation will allow you to take it off (we say the doughnut is "multiply connected").

Finally, the "2d Poincare conjecture" is the statement that the only simply connected closed 2d surface is the sphere. In other words, if you can't link a loop with your closed quilt then your quilt can be deformed to be a round sphere. A strong version of this was proved by Poincare, among others.

Now for the real "Poincare Conjecture" that was proved by Perelman, replace "2d" by "3d", so the quilt comes from stitching little cubes instead of little squares. The "closed and simply connected" assumptions are the same, and the conclusion is that the quilt is, up to deformation, the 3d sphere. It's much harder to visualize since now the quilt may not fit into regular 3d space. For example, the 3d sphere is what you get by stitching the whole boundary of the 3d cube together into one point (recall how we got a 2d sphere!) -- which is not something that fits into ordinary 3d space.

Is forming something like a Möbius strip or a Klein bottle allowed? Or am I thinking in a different direction?
I know a Klein bottle has weird characteristics; is it considered closed or simply connected?

Unfortunately, there is absolutely no way to describe this stuff in "human" terms, you really just have to get your head around the concepts and even then you are likely to have no idea what this stuff is on about. Mathematicians could spend their whole career not understanding this stuff, easily.

I'll try as best I can, but I can barely get my head around the most basic concepts here, so here I go: In topology we don't care so much about what you normally think of as mathematics, topology I guess you could

The/. eds could make this article 10x more relevant to most people by titling it 'Man wins million dollar mental masturbation prize' or by explaining the practical applications of this discovery. Instead the summary is a list of techno jargon that'd put Star Trek to shame with no mention of the $$ prize nor details of the winner. Who is this guy? Why did someone give him so much money for solving for x? Can I too win cash money for balls? If not, can I out source next year's winner to india and take a cut

OK you can behave this way, just so long as we're able to rudely dismiss as "balls" anything clever you ever do that is not immediately relevant to us. And your music collection and wardrobe and taste in partners too since we're on a roll.

most people on/. have no clue what this sentence means: Poincaré questioned whether there exists a method for determining whether a three-dimensional manifold is a spherical: is there a 3-manifold not homologous to the 3-sphere in which any loop can be gradually shrunk to a single point? The Poincaré conjecture is that there is no such 3-manifold, i.e. any boundless 3-manifold in which the condition holds is homeomorphic to the 3-sphere.

As someone who's job involves research into geometry and topology, I would like to point out that the summary is wrong in a couple of places. The Poincare conjecture states (in simple terms) that:

Any closed smooth three dimensional space ('manifold') without boundary where all loops can be contracted to a point is 'homeomorphic' (essentially the same as) the three dimensional sphere (that is, the unit sphere in 4 dimensions).

The words "homologous" and "boundless" have little/nothing to do with it.

Excuse me for replying to my own post. I should also mention that Poincaré's conjecture was not about 'a method for determining whether a three-dimensional manifold is a spherical'. It is simply the question of whether there are non-spheres in 3d which have all loops contractible (for a more accurate description, see the parent). The question about methods/algorithms for determining whether or not something is a 3-sphere is in itself very interesting though.

A triumph for Perelman. I hope he accepts the prize and rejoins the mathematical world. It is a little surprising that Hamilton did did share it as the Ricci flow was a crucial idea. But there is no doubting that Perelman did the heavy lifting.

For those of you who dismiss this result is of little worth, you will not likely see a comparable achievement of the human mind for 50 years.

Well, I think the real question in this case should be what is the topology of the shape in question (the human body)? Isn't the so-called "cavity" really just a long tube connecting two openings to the outer surface? If that be the only set of connected openings, then the body would be homeomorphic to a torus.

However, there's a complex set of connected openings in the head: 2 nostrils, 2 tear ducts, and the mouth all connect to each other inside. I don't know what this is referred to as, topologically.